Energy and Mass
Equivalence between Energy and Inertial Mass
Einstein's famous formula for the equivalence of energy and inertial mass appeared in Einstein (1905b) three months later than the famous (1905a) paper on the special theory of relativity. This equivalence is a consequence from the combination of kinematical and electrodynamical considerations. Let us look at the derivation of the formula, in a simpler form presented in Einstein (1946).
Suppose two inertial systems K and K', and assume that K' are moving with velocity v relative to K, as in the figure. Further, suppose a body B is at rest in K, and radiating complex S and S' (symmetrically positioned relative to B), each with energy E/2 approaches B and is absorbed eventually by B (S and S', like photons, have no inertial mass aside from the energy). Two sources S and S' are assumed because this way the body B remains at rest in K.
Now, seen in system K, two sources of radiation have momentum E/cc in total, according to Maxwell's electrodynamics, where c is the velocity of light in vacuum. But seen from system K' (with relative velocity v along y axis), the radiation makes an angle of α to x' axis, so that we have to calculate its contribution along y' axis, and this can be obtained by approximation, neglecting higher order terms, as is shown in the following figure.
Now, before absorption, body B has mass M, and its momentum seen from K' is Mv. The momentum of S and S' taken together, seen from K', is (Ev/cc); thus the total momentum is
Mv + (Ev/cc).
Then, suppose the mass of B is M' after absorption (that is, S and S' are both absorbed by B, so that M' must have increased, according to the the law of physics); then its momentum is M'v. Since the momentum must be conserved, in particular conserved along y' axis, we have
Mv + (Ev/cc) = M'v,
and hence M' - M = E/cc.
Thus, letting the increased mass m (= M' - M), we have E = mcc. (Both figures adapted from Einstein 1946.)
Maxwell's derivation of the velocity of light (and inclusion of optics into electromagnetism), as a consequence from his theory of electromagnetism, was a remarkable result, since almost no one then had noticed the relationship between light and electromagnetism. And Einstein's derivation of the equivalence of energy and inertial mass is no less remarkable, since kinematics and electromagnetic dynamics, then, seemed to have little relevance, if any, to the structure of atoms and the nuclear energy. Lise Meitner, one of the discoverers of nuclear fission, was struck by this formula when she first met Einstein in 1909; but she realized the full implication of this formula only in the winter of 1938, when she received a letter (in Stockholm) from Otto Hahn, the former collaborator in Germany. Enclosed with the letter was a carboncopy of the paper by Hahn and Strassmann, reporting a curious experimental result suggesting the occurrence of nuclear fission (see Rhodes 1986).
References
Einstein, A. (1905a) "Zur Elektrodynamik bewegter Koerper", Annalen der Physik 17 (1905), 891-921.
Einstein, A. (1905b) "Ist die Traegheit einers Koerpers von seinem Energieinhalt abhaengig?", Annalen der Physik 18 (1905), 639-641.
Einstein, A. (1946) "Elementary derivation of the equivalence of mass and energy", Technical Journal 5 (1946), 16-17.
[Japanese translation of these three, in 湯川秀樹監修『アインシュタイン選集1』共立、1971。]
Rhodes, R. (1986) The Making of the Atomic Bomb, Simon and Shuster, 1986, 253-262.
Last modified March 28, 2003. (c) Soshichi Uchii
